A large probability averaging theorem for the defocusing NLS

We consider the nonlinear Schrödinger equation on the one dimensional torus, with a defocusing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measures with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order , being the inverse of the temperature and a positive number (we prove ). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory. (paper)